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The Fencing Problem

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Introduction

Question: A farmer has exactly 1000 metres of fencing, with it she wishes to fence off a plot of level land. She is not concerned about the shape of the plot, but it must have a perimeter of 1000 metres. What she does wish to do is fence off the plot of land, which contains the maximum area. ...read more.

Middle

I made many more of these calculations and created a graph. This graph reinforces that 250 x 250 is the rectangle with the largest area which shows that the shapes with the largest area are those with equal sides. Number of Sides Area 3 48111.11 4 62500 5 68800 6 72168.72 7 73890.13 8 75444.17 9 76318.81 10 76942.09 Conclusion Regular polygons have larger areas than irregular polygons, so a rectangle could have the same perimeter as a square but it wouldn't have as much area. ...read more.

Conclusion

Because regular polygons with more sides have a larger area and a circle has an infinite number of sides a circle is the shape with the largest area. But because the relationship between number of sides and area is a curve rather than a straight line a circle has almost the same area as a 360 sided regular polygon or any other shape with a high number of sides. ...read more.

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